Packing six T-joins in plane graphs

TL;DR

This paper proves the conjecture for the case k=6, establishing that under certain parity and size conditions on T-cuts in a plane graph, there exist six edge-disjoint T-joins, advancing understanding of graph decompositions.

Contribution

The paper confirms the conjecture for k=6, providing a significant extension beyond previously proven cases and resolving an open problem in graph theory.

Findings

Confirmed the conjecture for k=6 in plane graphs

Established existence of six edge-disjoint T-joins under specified conditions

Extended the known cases of the conjecture beyond k=5

Abstract

Let G be a plane graph and T an even subset of its vertices. It has been conjectured that if all T-cuts of G have the same parity and the size of every T-cut is at least k, then G contains k edge-disjoint T-joins. The case k=3 is equivalent to the Four Color Theorem, and the cases k=4, which was conjectured by Seymour, and k=5 were proved by Guenin. We settle the next open case k=6.